{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# softmax"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-02-05T00:59:39.437329",
     "start_time": "2017-02-05T00:59:38.107280"
    },
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-02-05T00:59:39.445185",
     "start_time": "2017-02-05T00:59:39.439316"
    },
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def indentify(x):\n",
    "    return x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-02-05T01:09:22.474067",
     "start_time": "2017-02-05T01:09:21.627199"
    },
    "collapsed": false,
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[<matplotlib.axes._subplots.AxesSubplot object at 0x1130ac5f8>,\n",
       "        <matplotlib.axes._subplots.AxesSubplot object at 0x112392518>,\n",
       "        <matplotlib.axes._subplots.AxesSubplot object at 0x112349cc0>]], dtype=object)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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9Ozldqk0Se3WrIbMAJpzhOonEt5jVJ9UmAdjaspWeUI/G3yW5WDZ4u4D5h3eM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\nqAeArJQs5hfM58Z5NzJ/wnwWFCygZFy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      "text/plain": [
       "<matplotlib.figure.Figure at 0x112da3588>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# http://stackoverflow.com/questions/3985619/how-to-calculate-a-logistic-sigmoid-function-in-python\n",
    "from scipy import stats\n",
    "from scipy import special\n",
    "\n",
    "def sigmoid(x):\n",
    "    return 1 / (1 + np.exp(-x))\n",
    "\n",
    "pd.concat([\n",
    "    pd.Series(np.arange(-10,11), index=range(-10,11)).apply(sigmoid),\n",
    "    pd.Series(range(-10, 11), index=range(-10,11)).apply(stats.logistic.cdf),\n",
    "    pd.Series(range(-10, 11), index=range(-10,11)).apply(special.expit),\n",
    "], axis=1).plot(subplots=True, layout=(1,3), figsize=(15,5))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-02-05T01:21:10.389621",
     "start_time": "2017-02-05T01:21:10.378763"
    },
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# http://stackoverflow.com/questions/34968722/softmax-function-python\n",
    "def softmax(x):\n",
    "    return np.exp(x) / np.exp(x).sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-02-05T01:13:46.296393",
     "start_time": "2017-02-05T01:13:46.288477"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 0.01165623,  0.03168492,  0.08612854,  0.23412166,  0.63640865]), 1.0)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.array(list(range(-2, 3)))\n",
    "softmax(x), sum(softmax(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-02-05T00:43:57.105748",
     "start_time": "2017-02-05T00:43:57.102314"
    },
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# http://stackoverflow.com/questions/3985619/how-to-calculate-a-logistic-sigmoid-function-in-python\n",
    "# http://hamukazu.com/2015/07/31/mathematical-derivation-in-numerical-computation/"
   ]
  }
 ],
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